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To where did you go yesterday?
Where did you go to yesterday?
Where did you go yesterday?
I see people omit the 'to' a lot. However, I also asked a related question yesterday
Who did you give the keys to?
To whom did you give the keys?
Here, the 'to' seems necessary. Why can you omit the 'to' when talking about places?
Where has a double sense, both as an interrogative and as a relative. It may stand for either
a preposition phrase (PP) denoting a location or destination—approximately equivalent to at or in or to what/which place
Where are you? = At what/which place are you?
Where are you going? = To what/which place are you going?
But its pro-PP character is already less than obvious when it's used as a relative with a nominal antecedent.
The place where I am going ... = The place to which I am going ...
Moreover, there are instances where it is needed to designate an origin rather than a location or destination, and to avoid ambiguity where has to be coupled with a distinct preposition such as from.
I see where you're going, but I can't make out where you're coming from.
The first of these permits and the second requires that where should stand for
a nominal designating a place, which can act as the object of a preposition. The most formal style deprecates this pronominal use when a location or goal is designated. But this use is very common outside that style:
Come on, come on, lemme show you where it's at. —Dave Clark Five
Doesn't have a point of view, Knows not where he's going to —Beatles
And there's no evident reason in grammar for prohibiting where as object of some prepositions and not others. I myself follow the formal use (I actively avoid situations in which where must be employed with a preposition), and P.E.Dant speaks for his namesakes when he says that where ... to and where ... at are redundant; but those are stylistic judgments, not grammatical ones.
ADDED, with thanks to ruakh for raising the matter:
Note, however, that even in the most colloquial registers we avoid using locative prepositions like at and in immediately before where—*Let me show you at where it is— only when they are 'stranded' on the right side of the verb. Goal prepositions (to, by, until) and trajectory and origin prepositions (via, through, from) are borderline: they are apt to sound contrived in formal registers and pedantic in colloquial registers. My advice is that you avoid using where in situations where these constructions would be needed and instead use a frank noun: Which road did you take? or What countries are you going to?
Of your first three examples, only the second is correct:
Where did you go yesterday?
The preposition to is not omitted in this usage; rather, it is redundant and incorrect usage in the other two (although you will frequently hear "Where did you go to?" in colloquial American speech.) It is redundant because it is unnecessary: the meaning is clear without it.
In your other two examples, to is necessary because the meaning is not clear without it:
Who(m) did you give the keys?
